Space Travel - Lorentz Transformation

[Icheb]

Space ship Alpha travels at t = 0 and v = 4/5 c to the star Sirius which is 8.6 light years away. One year later spaceship Delta starts at v = 9/10 c to the same star.

Question 1:
When does Delta overtake Alpha, as measured from Alpha's, Delta's and Earth's perspective?

Question 2:
At which distance to Earth (measured from Earth's system) does this happen?

Regarding 1:
For Earth's perspective I use $$\delta t = \frac{1}{\sqrt{1-\beta^2}} * \delta t_0$$. Using that I get to 17.92 years for Alpha and 21.92 years (plus an additional year because the ship left a year later) for Delta. This would mean that from Earth's perspective Delta never passes Alpha.
For Alpha's perspective I use length contraction for the 8.6 light years and then use v = s/t to get the time for 4/5 c and the shortened distance. For Delta's perspective I use the same approach.

Would this be correct so far?

Now my question is regarding Alpha's speed as seen from Delta's perspective and vice versa. Would I just use the speed of the ship relative to the other for this and then also include length contraction? I'm not quite sure how to do this otherwise.

Regarding 2:
From Earth's perspective Delta never overtakes Alpha. In the other systems I guess Delta would pass Alpha. But since there's length contraction involved how would I approach this? Would I just calculate where that point is relative to the entire distance (meaning in %) and then apply that to the distance as seen by Earth?

 

[alphysicist]

Hi Icheb,

I think the speeds and distance measured in the problem are measured in Earth's frame of reference, so the times you find will not need to be transformed like you do in your post. Instead I think you'll find that Delta does overtake Alpha in Earth's frame of reference.

 

[Moetasim]

I can confirm that your approach to calculating the time for Delta to overtake Alpha from Earth's perspective is correct. However, you also need to take into consideration the time dilation effect for Delta due to its high velocity. This would result in a slightly longer time for Delta to overtake Alpha from Earth's perspective.

For Alpha's perspective, you are correct in using length contraction and the time dilation formula to calculate the time for Delta to overtake Alpha. For Delta's perspective, you would also need to take into account the length contraction and time dilation effects for Alpha. So both ships would see the other as moving slower and the distance between them as contracted.

In terms of calculating the speed of Alpha as seen from Delta's perspective and vice versa, you would indeed use the speed of one ship relative to the other and take into account length contraction. This would result in the speed being slightly different from what is seen from Earth's perspective.

For the second question, you could approach it by calculating the distance between the two ships at the point where Delta overtakes Alpha, and then using length contraction to determine the distance from Earth's perspective. This would give you the distance at which Delta overtakes Alpha as seen from Earth's perspective.

Overall, the Lorentz transformation is an important concept in understanding the effects of special relativity in space travel. It allows us to accurately calculate and understand the differences in time, distance, and speed between objects moving at high velocities.